An analysis of environmental factors affecting ice growth

TECHNICAL REPORT

AN ANALYSIS OF ENVIRONMENTAL
FACTORS AFFECTING ICE GROWTH

ELLIOTT B. CALLAWAY

Applied Oceanography Branch
Division of Oceanography

SEPTEMBER 1954

U. S. NAVY HYDROGRAPHIC OFFICE
(rs 2p, WASHINGTON, D. C.

ABSTRACT

This report describes efforts to evaluate the effects of
various oceanographic and meteorological variables on the
rate of sea ice growth, in an attempt to ascertain the most
important parameters to consider from the forecasting
standpoint. Work of Neuman and Stefan, as well as that of
Kolesnikov, is utilized in the form of prediction equations.
Results are applied to specific Arctic localities with good
agreement. It is concluded that the most important factors
affecting ice growth are air temperature, snow thickness,
and snow density, with several other variables playing a
lesser role.

Ol

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FOREWORD

The increasing importance of defense installations in
northern areas has greatly increased the responsibilities of
the U.S. Navy in supplying bases in Arctic waters, where sea
ice is often an operating obstacle. The Hydrographic Office
is charged with the responsibility of developing and testing
techniques for observing and forecasting sea ice conditions.
Standardized techniques for observing, charting, and reporting
sea ice are now in operational use by the Navy, as described
in publications issued by the Hydrographic Office. Heretofore,
techniques for forecasting the formation, growth, and movement
of sea ice have not been published by this Office. This publi-
cation describes in detail the tactors affecting ice formation

and growth.
It is requested that activities receiving this publication
forward their comments to the Hydrographic Office.

Charan —

J. B. COCHRAN
Captain, U.S. Navy

Hydrographer

|

O 0301

chet

DISTRIBUTION LIST

ONO (Op-03, 03D3, 31, 316, 32, 33, 332, Oh, 05, 533, 55)
BUAER (2)

BUSHIPS (2)

BUDOCKS (2)

ONR (Code 100, 102, 410, 416, 420, 430, nea 1,66)
NOL (2)

NEL (2)

NRL (2)

CQMOPDEVFOR (2)

COMSTS (2)

CODTMB (2)

AROWA

SUPNAVACAD (2)
NAVWARGOL (2)
NAVPOSTGRADSCOL, Monterey (2)
CCMDT COGARD (LIP) (2)
USC&GS (2)

CG USAF (AFOOP)

CGAWS (2)

CGNEAC (2)

USAF CAMBRSCHLAB (2)
USWB (2)

CIA (2)

BEB (2)

SIPRE (2)

ASTIA (2)

ARTRANSCORP

CE (2)

INTIHYDROBU, Monaco (2)
ARCRSCHLAB, COL, Alaska
ARCINSTNA (2)

WHOT (2)

SIO (2)

UNIV WASH (2)

TEXAS A&M (2)

CBI (2)

iv

CONTENTS

Forewo rd e e e e e e e e e e e e e e e

DUsoributvon Wise es soso ee senile.

List of Figures OO ee O20 je. Henge) <2: 26) ae

Ae

Be

Bibliography e e e e e eo ° e e e e e e oe

MnteroGUCt VOM ve ie)iel toncelare. ve ueliel Jello

Increase of Thickness with Time. .

Kolesnikov's Equation and its Evaluation... .

Physical Constants used in Numerical Evaluation.

e

Effects of Changes in Meteorological and Oceanographic
Parameters on Ice Thickness. .....o «

Practical Ice Thickness Forecasting. . ....

Empirical Formulas for Ice Growth Prediction .

Coneriuistion: <6) «1 lore elle Menton tone

V

OF 50), MOL Ne. NOAes Ke)

Figure

Figure

Figure

Figure
Figure

Figure

Figure
Figure
Figure
Figure
Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

17

18

Ww)

LIST OF FIGURES

Graphical Solution of Equation 19. « . « o » « «6

Equivalent Temperature minus Air Temperature as a
Function of Méteorological Variables. . . « » ° 6

Influence of Meteorological Conditions on the |
Freezing of Ice. Dee at BOONE Ci CIOL OS Oi

Effect
Effect

Effect
on Ice

Effect
Effect
Effect
Effect
Effect
Ef fect
Effect

Effect

of Snow Depth on fee Growth. eco ec oe ee
of Snow Tee on Ice Eels 8.0 88

of Equivalent Temperature and Snow Depth
Growth e i) ec ® ° ® e © ® ry e ® ‘° @ @ e @ Ry 6

of Initial Ice Thickness on Ice Growth. .

eee

snes
- ; 4 i ADS

eo L7

2 e 17

of Snow and Ice Thickness on Rate of Freezing.. 19

of Wind Velocity on Ice Growth. o » » o « e
of Salinity on Rate of Freezing. 0 9 @ @ 6
of Heat of Water Mass on Rate of Freezing...

of Water Temperature on the Growth of Ice.e

oo 19

wae

eo cas
eo 22

of Thickness of Mixed Layer on Growth of Ice. « 22

of Cloudiness and Wind Speed on Growth of

Ice (Freezing Time 14 hours) oo Cuomo owen (Od O6 Omar

Effect

of Cloudiness and Wind Speed on Growth of

Ice (Freezing Time 14 hours) «e SEO oO 60 0 0-010 6 6 Bb

Observed and Computed Ice Growth at Archangel, U.S.S.R.26

Observed and Computed Ice Growth, Denmark Expedition
Records, (Nov, 1, 1906; to Feb. 6, 190/26. 6 os ee eric

Observed and Computed Ice Growth at Padloping Island,
NieWeliers) CanuUarys I OLO seu: tema Mieiilen aii elion ee em ciialontei emer

Observed and Computed Ice Growth at Padloping Island,
N.W.T.; January WOOSG) 6°06 .0 606 066.0600 00 BH

Empirical Formulae of Zubov's Type Compared with
Observed Ice Growth at Archangel, U.S.S.R. « « « « © e 29

vi

A. INTRODUCTION

Forecasting ice thickness can be separated into three problems:
(1) determining the time required to reduce the temperature of the
water mass to the freezing point by thermohaline convection; (2) fore-
casting the temperature, wind velocity, cloudiness, humidity, depth,
and density of snow fall for the period for which the ice thickness
| forecast is required; and (3) computing the thickness of the ice accre-
tion which will result from the predicted weather conditions. This
study is restricted to the last phase of this problem.

The problem of ice thickness forecasting is*one of great complexity,
expressible only in terms of a complicated system of differential and
integral equations, the solution of which 1s not possible when the bound-
ary conditions are not simple. The first physicist to present a complete
mathematical theory of heat conduction was Joseph Fourier. The applica-
tion of Fourier's heat conduction equations to the problem of ice forma-
tion was first undertaken by Franz Neumann (Weber, 1910) and Stefan (1889).
More recently Russian and Norwegian scientists have been active in this
field, and it is with the description of the work of A. G. Kolesnikov

(1946) and Olav Devik (1931) that this study is chiefly concerned.

After the derivation of a theoretical forecast formula the question
arises as to the best practical method of its application. In this con=-
nection consideration should be given not only to the facility with which
the results of the formula can be obtained but also to the accuracy of
the evaluation.

There are two general methods for obtaining the required ice thick-
ness from the formula: (1) by computation for each individual situation,
and (2) by taking the required hickness from graphs consisting of para-
metric curves of ice thickness drawn with temperature and ice thickness
as abscissa and ordinate, respectively, and with meteorological factors
as parameters. Both of these methods will be derived and explained. To
quickly obtain approximate results the graphical method is recommended,
but for a more accurate determination in which all the parameters are
given individual consideration, the computational method should be employed.

B. INCREASE OF ICE THICKNESS WITH TIME

In approaching the problem of ice growth with time, it is desirable,
initially, to formulate an expression in simple terms, that is, in the
preliminary steps to neglect the meteorological factors of wind velocity,
cloud cover, and humidity and to consider the ice as formed free from the
blanketing influence of a snow. cover. Initially then, assume a surface
of still water lowered by contact with air to some temperature To below
the freezing point. There will then be formed a layer of ice whose thick-
ness ¢ is a function of the time t. A solution may be reached by equating
the amount of heat carried up from the water below the ice sheet plus the
heat set free per unit of time (dt) per unit area (as the ice increases
in thickness by dé) to the total amount of heat that flows outward through

a unit area of the lower surface of the ice sheet. This heat puceet

equation must satisfy Fourier's heat transfer equations:
2

OT, 3 0 it in the ice (O<x< €) ay
Ot or dx* gp
Ole gale fe : yt
Oe] 2S S ote —— in the water (E<x) i (2)
Ot Ox | ,

where:
Tj = temperature in the ice,

To = temperature in the water,

Q,= ENS is thermal diffusivity in ice,
Cy, Pi
pe 2 is thermal diffusivity in water, ,
Qa = 2 Pe
t = time

k, = thermal conductivity in ice,
kg = thermal conductivity in water,

cy = specific heat of ice,
cg = specific heat of water,
Pj = density of ice, and

P25 density of water,

The temperature of the boundary surface of ice and water (at x = é)
must always be O°C (under this simplified formulation) and there will be
continual formation of new ice. If the thickness increased by dé in time —
dt, there will be set free for each unit of area an amount of heat © .

a= Lp, nt sb mena

where L is the latent heat of fusion. This heat must escape upward by
conduction through the ice, and in addition heat must be carried away

from the water below, so that the total amount of heat that flows out—
ward through a unit area of the lower surface of the ice sheet is

aan ‘ ui} ; ray He
Q=k ( | at ee
of this amount the quantity Pics
qi. k ( Te ) dt re (5)
x /x=e

flows up from the water below. Hence the first boundary condition is

(x, 21 _ 4, Ste) Se Mpoleiiite (6)

Ox Ox / xzé Ot

The other boundary conditions are:
T) =T,=C, at x=0, : (7)
Ty) =T,70 atx =¢, and (8)
To = Co at x = ©, (9)

There are also three other boundary conditions derived from the fact

that when t = 0, € is fixed, while T) and Tg must be given as functions
of x. T) lies between O and € while Tp lies between ¢ and ™. As
equation (6), containing the unknown function €, is not linear and
homogeneous, a solution cannot be reached by the combination of special
solutions. The method of solution then will be to find particular inte-
grals of equations (1) and (2) and after modifying them to fit boundary
conditions equations (7), (&), and (9) to find under What conditions the
solution will satisfy equation (6). This will also determine the initial
values of ¢ , Tj and To.

Now the function @(x,7)(the probability integral) is a solution of
such differential equations as (1) and (2). Consequently if By, Dj, Bo and

D2, are constants and 1» = 7 Yast lil Gg Say Wane | 6

T, = B+ Di b (x,%) (10)

and

1b >) Bo HD p Dy (xno) (12)

are also solutions. Boundary condition (8) means that ®(é>» ") and
@(é,7,) must each be constant, which will be true if £=0, g=m,
or if €° is proportional to Vt. The first two of these assumptions
are evidently inconsistent with (8). Thus, there remains only the last
which may be put into the form

& = sa oe)

where b is a constant to be determined, together with By» Di» Bo, and Do. .

From the properties of ® (X)it is known that @(0) = 0 and (GC) B 1k
Fitting boundary conditions (7), (8), and (9) in (10) and (11), with the
use of (12), the following equations result:

et OD se
b ,
By+ 0 (2. )=0, (14)
i \ 20/0
b \=0, and (15)
B,+ D, ® ( se ,
Sy ai Wee as Arne
while (10), (11), and (12) in connection with (6) give ;
bis b2. whe
Me Di @- 4, a he Danone, ve odkdide, a0
f~W7ar, t {17st 2 il :
Solving equations (13) to (6) for Dy and Do yields
Ci ae C2
Oi Fa e/a ’ CaS b ’
(i) Ge)
2 ES. 2 a, o ;
and substituting these values in (17) finally gives
er bos Ebina
sd a
2/L | NO,

This derivation is due to Neumann.

This transcendental equation (19) can be solved for b by plotting
the curves

SE ot and Y= f(b) ,

where f(b) represents the left hand side of (19). Then b is found as
the abscissa of the intersection of the two curves. ; : by

Figure 1 shows a graphical solution of (19) by this method, using
the following values for the constants involved:

GRAPHICAL SOLUTION OF EQUATION 19.

I.

FIG.

= ,0053
cl = 20°C.
oC, 0118
kg = .00143
co = -1°C.
= .00143
a = 80.0
1 = 0.92.

y
rah
H

Using the value of b found from Figure 1, the equation relating the
thickness of ice to the time, for an ice surface temperature of -20°C.,
and water temperature of -1°C., is

€=0.0531,/t i | ees)

where £ = ice thickness in cm. and t = time in seconds. An evaluation
of (20) for 14 hours gives an ice thickness of 12 cm. as compared to

8 cm. from the practical forecast curves of Figure 12. The differences
may be due to the different values of constants used, i.e. V = 20, N = 0.

Stefan in a similar fashion derived an expression for the ice thick-
ness as a function of time. He made a further simplification by assuming ~
the water temperature to be 0°C. This is merely a special case of
Neumann's solution and can be obtained from it by making Co = O in equation
(19). To a first approximation Stefan's equation is

2
Bie = Ey Orn (21)
L

€ = ice thickness,

C] = temperature of the ice surface,
L = latent heat of fusion,

¢, @ specific heat of ice,

thermal diffusivity of ice, and
tine.

where

cr
ow

(cc, is available from tables but is equal to the thermal conductivity
divided by the product of specific heat times density--all available
from tables).
Equations (19) and (21) are not suited for practical.application,
since the effects of initial thickness of ice, thickness and- density of
snow cover, wind velocity, cloud cover, humidity of the air, and, salinity
of sea water are neglected, and in addition such a long and tedious. com-
putation is not suited for practical use. To take the meteorological —
conditions into account, it is necessary to form heat budget equations.

Consider first the sunlight and daylight radiation. At a low sun
altitude of from 5 to 10 degrees, for example, a water or ice surface
will reflect a considerable part of the sun's rays. A fraction will
enter the water and be gradually absorbed, while the transmitted ra-
diation will be practically all absorbed by the lower layers of the
water body. At a sun altitude of 5° one square centimeter of water
surface will receive 0.6 calorie per square centimeter per hour. This
is only 0.5 percent of the amount of solar radiation reaching the atmos—-
phere.

Under these conditions and with a clear sky, the diffuse daylight
from all parts of the sky will give a greater input, namely 3 cal/cm$
per hour. If the sky is clouded, the diffuse daylight absorped by a
water surface will be less but will still represent 1 cal/emS/hr, which
is more than the direct radiation would give at the low sun altitude.

These figures of heat gained are small compared to heat losses in
winter. But when the sun's altitude increases, the incoming radiation
direct and diffused will contribute more and more to the heat absorbed;
and in the summer the relation will be reversed, as the heat received
will exceed the heat lost.

Restricting the problem to winter conditions, next consider the loss
of heat by infrared radiation at a representative air temperature of -10°C.
With a clear sky and air temperature -10°C. the heat radiation from a
water surface at O°C. will be about 1, cal/cem/hr.

Nearly all surfaces will absorb almost completely infrared radiation
of the type radiated from objects of moderate temperature, and with re-
spect to these rays,water is a black body. This i8 also the case with
snow. This means that infrared radiation falling on a water or snow sur-
face is completely absorbed in the uppermost layers in a very thin sheet
some hundredths of a millimeter thick. On the other hand the same layer
emits infrared radiation of the same wavelengths to the atmosphere,

The next process to consider is the loss of heat by convection.
This loss is due to the difference in temperature between the water
surface and the air above, but it is also modified by the wind. It is
well known that a body is much more rapidly cooled when a wind is blow-
ing than when it is calm. The figures relating to a water surface are:
heat loss by convection 2.8 cal/em?/hr, with an air temperature of -10°C.
and calm air, and heat loss by convection 11.5 cal/em2/hr, with a wind of
5 meters per second.

Another heat loss to be considered is the loss caused by evaporation.
The evaporation depends first upon the dryness of the air above the water
surface, or more exactly upon the vapor pressure, and secondly upon the
wind. With an air temperature of -10°C., a vapor pressure of 3.5 milli-
bars, and calm air, the loss by evaporation from a water surface at O°C.
is 1.7 cal./cm.“/hr., but if there is a wind of 5 meters per second it

will increase to 7.7 cal./cm.“/hr.

Adding all the heat losses and subtracting the heat gain, the net ~
loss under the first set of meteorological conditions, i.e., air temper= .
ature -10°C., vapor pressure 3.5 mb., wind velocity 0 meters/second, . °~
cloudiness 0%, sun altitude 5°, and water temperature O°C., will be
14.7 cal/cm2/hr. In the second set of meteorological conditions, i.e.)
air temperature -10°C., vapor pressure 3.5 mb., wind velocity 5 meters/'
seconds, cloudiness 100% (overcast), sun altitude 5° and water temperature:
0°C., the net loss will be 23.1 cal/em?/nr. These figures are equivalent.
to the production per hour of a sheet of ice of thickness 2.0 and 3.2 mm.»
respectively.

The above briefly outlined analysis forms the basis for the calcu- »
lation of the actual growth of an sheet ice under conditions existing in
nature. The method could be applied directly if the surface of the ice.
retained the temperature of O°C. and had the same physical properties as
water with regard to heat exchange. The first condition is not true, how-
ever, as the surface temperature of the ice will be lower when the ice is
thicker, if the air temperature is low. i

The flow of heat through the sheet ice takes place in accordance with
Fourier's heat conduction equation (1), and the heat budget equations out—”
lined above must be solved in accordance with this equation.

C. KOLESNIKOV'S EQUATION AND ITS EVALUATION

Kolesnikov has recently (1946) derived an expression for the thick-
ness of ice as a function of time which involves all of the meteorological
factors. This was accomplished by setting up heat budget equations in-
volving these parameters and solving in connection with Fourier's heat
conduction equation.

The form of Kolesnikov's equation for salt water is
Abs | SiS + ary eee eee ee

i+ 1 "Ba fs ea cw ilNea8 a
"Seo ee ee ea

where ae initial ice thickness in cm.,
= increase in ice thickness in cm.,
S] = salt content of the ice,

= salinity of salt water solution at temperature of freezing,
Bix
f, = density of ice, 0
FP:
Kt: K+T (Coz -C,),

density of sea water

snow thickness in cm.,

snow density,

wind velocity in m/sec.,

air temperature in °C,

C, = specific heat of sea ice,

T,, = temperature of freezing in °C.,

© = equivalent temperature in °C,

Co = specific heat of sea water, and Si

K = latent heat of crystallization of salt water ice Bollea)

yd
ESO M% pd
uw 08

The equivalent temperature © is a resultant temperature which takes into
account the effects of (1) radiation, (2) convection, (3) evaporation
and condensation, (4) wind velocity, (5) humidity, (6) cloud cover, and
(7) insolation. It is defined by the following relation

GO = +) a (22a)

where ©= equivalent temperature in °C.,
T= air temperature in °C.,

q = total heat exchange represented by the seven meteorological
factors enumerated above,

er Cerny ty Ci 5

C8 Bios) 2s 10714 x Be,
Vo = wind velocity in m/sec.

The total heat loss q is determihed as the sum of the individual
heat exchanges as follows:

=_ -.06 P
radiation loss = 1.307 x 107)? x 7, (0.255 + 0.322 x 107°°°7P0)(3 - e.ng),
convection loss = 1,75 x ¥ 902858 x lo-4,

evaporation-condensation = 145.h4(ew - ea)Wa/ 140,
insolation: taken from Meteorological Tables,
where
Po = vapor pressure in mbs.,
Cy = cloud coefficient defined as follows:
0&6 for Nb, St, and Sc,
e77 for Ac, and
020 for Cse,
n, = cloud cover in tenths,
@€, = vapor pressure in inches,
e, = 0.98 e,, and
Wa = wind velocity in knots.

The combined effect of various meteorological factors on the ice thiek-
ness is shown in Figure 2. Approximate average values of ey (is for”
Arctic latitudes are listed in Table I. These values, in the’ frur of
heat gain and loss, are tabulated by months during which | ice formation

occurs. ete ee

TABLE I site.
Heat Gain or Loss (24/a,) ane? =
for the Arctic Regions (ay f oe iene BE SS
Month Heat Loss Heat Gain Net Gain (4).or Lossi{=
Nov. 5 02°C ° 0.2°C ° =5 200°C e ;
Dec. 5 4C. Os26°C% =5 2h 5 eon ese) 30.7
Jane =5 9%. Oh49°C 6 : =5Sehl Ce” These 3B
Feb. —6,0°C,. 1.85%. el FC 8h a 2
Mar. —6.5%CT., 5.95. =0.55°C we 628!
Apr. —6.6°C. 9.60°C. $3.00°C .

By the proper choice of parameters and physical constants, the
following equation for the practical coe ciananeler, of ice eee tn:
fresh water is derived: i

8 100 heed ape
h, + [1.49954 - Pe h
p? 1.75 a as 5.23 X10 andes eee

(23)

9
= 12), 1+0.00330 ©

where h) = increase in ice thickness in cm,

snow thickness in cm,

7D
°
i

= snow density,

wind velocity in knots,

air temperature in °C., ay: iG
initial ice thickness in cm, and ee aah
§ = equivalent temperature in °C.

fe]
°
toa

a
°
W

While equation (23) looks complicated and clumsy, it can be evaluated
easily by substituting the given meteorological parameters into the
numerator and denominator and dividing.

10

TIME (DAYS)

TIME NECESSARY TO

FREEZE 5cm. OF ICE

UNDER VARIOUS METE-

OROLOGICAL, CONDITIONS,
&=2cm. A & = 5cem, NO

SNOW.

RELATIVE HUMIDITY (R.H.)

Ty °C -5
Vp (KNOTS) 2
No (% CLOUDINESS) 80
RY (%) 20

FIG. 2.

-10 -15 -20 -25
4 6 8 10
60 40 20 {0}
40 60 80 100

INFLUENCE OF METEOROLOGICAL CONDITIONS ON

THE FREEZING OF ICE.

11

D. PHYSICAL CONSTANTS USED IN NUMERICAL EVALUATION

In the practical numerical evaluation of Kolesnikov's formula (22)
for a given set of meteorological conditions, the following values for
the physical constants involved were used.

l. Density of Sea Ice

Utilizing the investigations of Malmgren.(1927), which indicate
the independence of the ice density from the Woman: the value of the
ice density was taken as © = 0.916.

2. Latent Heat of Fusion

The latent heat of fusion depends upon the taal content of the ice.
Malmgren's investigations give the following value for, ‘K, the latent heat

of fusions: S i
ee

where S, is the salinity of the ice and ois the sarintty of the sea
water. The formula indicates that the-value of K decreases with inc rease
in salinity and increases with increase of sea water salinity for a
given salinity of ice. re Ee s

3. Specific Heat of Sea Ice (01).

The specific heat of sea ice is dependent upon the salinity. How-
ever, this dependency is well marked only for smal) negative temperatures,
i.e., in the vicinity of O°C. For large negative temperatures the de-
pendency of the specific ‘heat upon salinity is greatly decreased.
Malmgren's observations indicate that after the initial stages of ice
fomation, for temperatures between -8°and -14°C. and for salinities of
from 4 to 6 °/oo, the specific heat varies between 0.57 and 0.88. There-
fore, by using small time intervals in the forecast, the Wego heat may
be considered as a constant.

4. Thermal Conductivity of Sea Ice (A4) Me vi

As between the one actually observed determination of the iiemal
conductivity of sea ice by Malmgren of .0051 cal/cm?/sec. and the value
of .0045 found by three Russian scientists, Kolesnikov considers this
latter value the more reliable.

5. Thermal Conductivity of Snow (X,)

Abels' (1892) formula for the thermal conductivity of snow is used:

Ag = 0.0068 42

where “, is the density of snow.
6. Convective Heat Loss
For the coefficient of heat loss through convection, Frank's
(1929) formula is used:
0.656 SC
a. = 1.75 X Vo xX 10 ;
where Vo is the wind velocity in m/sec.

7. Density of Sea Water

For purposes of evaluating the formula, sea water density is con-
sidered as constant and equal to 1.000 throughout the whole period.

8. Specific Heat of Sea Water

As it varies but little with salinity, the specific heat of sea
water (Cj) is considered a constant and equal to 0.975.

9. Effective Radiation of a Black Surface

From Angstrom's formula for the effective radiation of a black
body

12 4 -0.069P
Re,='376X10 x To {0.255 + 0.322 x 10 of ;

where T, = air temperature and
Po

Taking from Falkenberg (1928), the emissivity of the snow, a, as 0.995, it
follows that

pressure of water vapor in mb.

-/2 4 -0.069P,
aRe, =1.307X10 X To 40.255 + 0.322 X IO

Therefore the heat lost through radiation is
Gin 5623 x 10) te

Devik (1931) found by means of the above formula that for air temperatures
between O and -20°C. and for relative humidities close to saturation,
the magnitude of aRe, does not vary much. Therefore it can be taken

as a constant equal to 30.6 x 107.
10. Effect of Cloudiness on Radiation

This is calculated by means of the following relation:
aRe = aReg x (1 - econo),
where Cy is a coefficient taking into account the diminution

13

in the total radiation due to the mean cloudiness, and no is the average
cloudiness in percentage. Assigned values of Co are 86 for nimbostratus,
stratus and stratocumulus clouds, .77 for altocumulus, and .20 for
cirrostratus clouds, from the study of Efimov (1939). Ths reflection
coefficient is taken as a constant equal to 0.65.

E, EFFECT OF CHANGES IN METEOROLOGICAL AND OCEANOGRAPHIC PARAMETERS ON
ICE THICKNESS

1. Snow Thickness

The factor having the most pronounced influence on the rate of ice
growth is the blanketing effect of a layer of snow on the ice surface.
A quantitative graphical evaluation of this effect is presented in Figure
3. The parameters have been given the following values:

Initial ice thickness (3) = 15 cm.

Snow density ( “ ) = 0.3

Wind velocity (vo) = 4.5 m/sec.

Air temperature (To) = -20°C.
Equivalent temperature (©) = -25°C,

The rate of growth of the ice for snow thickness § of 0, 10, 20, 30, 40,
and 50 cm. is indicated. The following facts are evident from the figure:

The most rapid ice growth takes place with an ice surface free from
snow. For this surface the most rapid growth takes place with tne lesser
ice thicknesses. Here a decreasing rate with increasing ice thickness is
evident. Upon reaching a thickness of approximately 50 cm. the rate of
growth becomes essentially linear but at a much lower rate than in the
initial stages, To add this 50 cm. of ice requires about 18 days with an
average temperature of -20°C. From this point on, the rate of growth is
essentially linear at the rate of about 10 cm, in & daya.

A snow layer only 10 cm. thick makes a marked change in the charac-
teristics of the rate of growth. To reach a thickness of 50 cm. now takes
about 48 days. The ice thickness increase is essentially linear with time,
requiring approximately 8 days to produce an increase of 10 em. in thick-
ness,

With increasing thickness of snow, the ice thickness increase remains
essentially linear but with progressively decreasihg slope; with 20 cm. of
snow it requires about 14 days to add a thickness of 10 cm.3 with 40 cm.
of snow about 20 days are required to add 10 cm.; and with 50 cm. of snow
about 32 days are needed to add 10 cm, to the ice thickness.

14

LE (cm)

TIME (DAYS)

oo
u
e

80 T hast poe

€,= 15cm.
B= 0.3

iH

70)

Vo = 1Omi/hr

@ii-25ec

Tp =- 20°C

5 = SNOW DEPTH
oe

?

UNERU Tae

|
bans

Ee
aa

We eh

N
N\\

ne) 4 3 12 16 20 24 28 32 36
TIME (DAYS)
FIG.3. EFFECT OF SNOW DEPTH ON ICE GROWTH,
36
tg2 TIME DUE TO SNOW
2, Aێ= ICE THICKNESS INCREMENT
32 YT, ona $= SNOW THIGKNESS
e Po: SNOW DENSITY
4 @= EQUIVALENT TEMPERATURE
28 oe [ TOTAL TIME FROM EQUATION (23)
on INITIAL ICE THICKNESS = 45.7 cm.
0, 8=7.6 cm.
Agé=22.8 cm.
oe £0 1.498
Ne; Ag (aa?
eo
y ts=
A
20} NZ, 1 12 Dtso.00380
oe
Xe
16 aN T le
(e)
Sy
{o)
& |
12 e
7
8 + i=
4
0.2 0.3 0.4 0.5 0.6 0.7 0.8

SNOW DENSITY
FIG. 4, EFFECT OF SNOW DENSITY ON ICE GROWTH
15

WARS

b

i

0

2. Snow density

The factor next in importance to snow thickness in determining ice
thickness growth with time is snow density. In Figure 4 the initial
jee thickness is 45.7 cma, the ice increment 22.8 em., snow thickness
7.6 cm. and snow density varies from 0.2 to 0.8. Curve (1) of Figure
4& shows the time required to add this ice thickness increment of 22,8
em. with varying snow density. The time diminishes rapidly at a non=
linear rate up to a density of 0.5, where it becomes more nearly linear
and diminishes slowly up to the maximum density of 0.8. In going from
a density of 0.2 to one of 0.5 the time diminishes from about 35 days
to 16 days to add on the ice increment of 22.8 cm. When the density
changes from 0.5 to 0.8, the time drops only about 2 days, from 16 to 14
Cays.

Curve (2) shows the amount of time that is added by the snow layer
alone. This curve shows that the snow contributes a much higher per-
centage of the total time at low snow densities than at high densities.
At low snow densities the snow layer contributes about 63% of the total
time while at high snow densities it contributes only about 10% of the
total time.

3. Air and Equivalent Temperatures

The next meteorological factor to consider is O=lo+Z Nees which
is the sum of the air temperature and the net temperature change due to
the heat exchange at the surface of the snow or ice. The quantity = Ve
is analyzed and derived above. Figure 5 shows the effect of a change
in ©, both for ice formation in salt water (curve 1) (equation 22) and
in fresh water (curve 2). Curve 3 indicates the change in time for ice
growth in fresh water due to changes in snow thickness, The vertical
scale indicates the time in days required to add ice thickness ihcrements
of 2 centimeters.

Curve 1 indicates the nonlinear nature of the variation of time with
ice thickness for different values of snow thickness on salt water ice.
The slope shows a definite increase with increasing thickness of ice.

It requires approximately 35 days to add 14 cm. cf ice under the indicated
conditions with © -= -10°C., and approximately 12 days with © = -30°C.
Curve 2 indicates the variation for fresh water under the same conditions.
It is seen that the growth of ice in fresh water reouires slightly less
time than in salt water under the same conditions, Curve 3 indicates the
variation in time for ice growth in fresh water for different snow thick-
nesses.

4. Initial Ice Thickness
The next parameter to consider is that of initial ice thickness (é, No
Figure 6 shows that the variation in ice thickness growth with time is

linear for constant values of initial ice thickness. Greater values of
ice increment result from lower initial ice thicknesses, The relative

16

TIME (DAYS)

ICE GROWTH IN Gm. (A €)

hy CURVE | CURVE 2 CURVE 3
32 SALT ‘FRESH FRESH
WATER WATER WATER
2 Aé-2
30 53-65 €, -53 to 65
oO -11 §-0 tol! a
28 02 & -0.2 oS)
| 6.5 mishr. Vg -6,.5 mi/hr ce)
-1,.8°C Sq,-FRESH WATER =“ (
26'— 0.916 ~——P,--0.916 gn
| 1,00 P, -FRESH WATER |,
24/— -20°¢ = Tp -20°C sb
ln OL7215 c, -0.725 aw
22|-— -1.68°Cc Ty -FRESH WATER =x
~10,-20,-30 @- -10,-20,-30
20|_ 0.975 C2 -FRESH WATER
66.28 -FRESH WATER

8 10 14
ICE GROWTH INem (A€)

FIG, 5. EFFECT OF EQUIVALENT TEMPERATURE AND SNOW DEPTH ON
ICE GROWTH.

R= 0.2

- Voz 10 Mi, PER HOUR
ta To? - 20°C,
@=- 25°C,

10 12 14 16 18 20

TIME (DAYS)

FIG, 6 EFFECT OF INITIAL ICE THICKNESS ON IGE GROWTH.
17

importance of initial ice thickness and snow cover thickness ia clearly
shown in Figure 7, where curve 1 indicates the variation in time to
freeze 5 cm. of ice for different initial ice thicknesses and curve 2,
the time required to freeze the same amount of ice with varying snow
thicknesses.

5 « Wind

Figure 8 shows the effect of wind velocities of 20, 30, and 40
miles aBer hour on rate of ice growth forA€ = 1 cm, f4 = 53 cm. and
10 em. Under these conditions a change of 10 miles per hour in
wind mleeasy has very little effect on ice growth. However, for lower
values of wind velocity the effect is marked

6. Salinity

The effect of sea water salinity is indicated in Figure 9 Under
the conditions indicated on the figure a slightly greater ice thickness
accumulates for the length of time in fresh water than accumulates in
salt water. However, the two curves do not show much difference in
growth rate of ice due to the salinity.

7. Effect of the Heat of the Water Mass

None of the formulas so far considered has attempted to evaluate the
effect of the heat of the water mass itself on the growtn of the ice
sheet. In the presence of warm currents or where the thickness of the
layer subject to convectional cooling is great, the effect of the heat
of the water mass is appreciable. By taking the heat of the water mass
into consideration, a formula can be developed which is identical with
Equation 22 with the exception of an additional term which represents
the heat of the water mass.

The form of this additional term is as follows:

= x 2nel ene
t62C, pa(T,-T))H,, @ expl-A,F (2*o a | exon H 4,1

c - 2 2
Keliscr(m@] | mI (2-1) eS
where: C2 = specific heat of sea water,

ee) = density of sea water,

= average temperature of the layer in °C,
T> = freezing point of sea water in °C,
= thickness of layer expressed in meters,
= density of ice,
C, = specific heat of ice,
= equivalent temperature in °C,
Ap = turbulent heat conductivity of sea water,

18

(CE GROWTH IN cm.(A€)

20

tina are

TIME (DAYS) TO FREEZE 5 em. ICE

FOR VARIOUS €, AND 8 VALUES |
&=INITIAL ICE THICKNESS (cm)

8=SNOW DEPTH (cm) TI i" ry >
To=-25°C. I L
He |

18

16 Vo=l0 m | cf |
R.H=80 % RS
CLOUDINESS=0 %

Ry
LHP LL Cee
Waoeeels
BEUEOIEOTONERREGI AI

r
{ LJ le
voll LH i PEAT

TIME (DAYS )

i
: AUASEAHEGALEGALLE
CTT tt tt He a

.) acl ICE THICKNESS (&)

—

fo} pens 2
cae 6 10 14 18 22 26 30 34
5 10 20 25 30 35

iCE THICKNESS AND SNOW DEPTH (cm)

aera 7. Nee coe OF SNOW AND ICE THICKNESS ON RATE OF
EEZ

| (1) WIND VELOCITY = 40 Mi per hour
(2) Bee

(3) =20
8)-——— 8= 10cm. inall cases
€ = 53cm,

7 eee

TIME (DAYS)
FIG, 8.EFFECT OF WIND VELOCITY ON ICE GROWTH
19

From this equation, it appears that the larger the depth H to which winter
vertical convection reaches, the higher the mean temperature of the layer,
and the smaller the coefficient of turbulent heat conductivity, the more
important this term becomes. The relative effect of the heat of the

water mass is decreased by an increase in the thickness of the snow and
ice layers. This effect could be foreseen intuitively, since both the

ice and snow layers act as excellent insulators and considerably decrease
the flow of heat from the water mass outwards, thus keeping the latter
from cooling.

- Taking a resultant temperature of -20°C., the freezing point of sea
water (T, ) as -1.8°C., the average temperature of the layer To as 0.5°C.
and the turbulent heat conductivity of sea water (A2) as 8, the following
expression results:

-8.47 x10" (&)2 (2n-1)?

y 2
2.76 amen recrs sane (25)

The evaluation of the infinite series of (25) is not particularly dif-
-ficult, as it is rarely necessary to carry the computation to more than
3 or & terms to secure the required degree of accuracy. Carrying out
the evaluation of equation (2h) by using the parameter values as in equa-
tion (25) and miltiplying by Ag and dividing by the right hand side of
equation (22) evaluated for an air temperature of ~20°C. yhelds the re~
sults shown in Figure 10, where the horizontal scale is in days of
freezing time added for an air temperature of ~20°C., and the vertical
scale is ice thickness in cm. Each isoline of mean water temperature
indicates the freezing time added by mean temperatures ranging from
-1.5° to 0.5°C. in the water layer, which is assumed to be 100 m. thick,
and the freezing point of sea water T;, is taken as -1.8°C. The figure
shows that the heat of the water mass adds to the freezing time least
when the water temperature is lowest. At 4 mean temperature of -1.5°C.
the heat of the water mass increases the time for freezing 10 cm. of ice
from an initial thickness of 15 cm. by about 1.1 days, and to the time
for adding a thickness of 70 cm. to an initial thickness of 15 cm. by
about 7.6 days. The variation between these two points is essentially
linear. At the warmest mean water temperature of 0.5°C., the heat of
the water mass adds about 10 days for the 10 cm. addition (to the initial
thickness of 15 cm.) and about 54 days for the 0 cm. increment of ice
thickness. At this water temperature the variation is nonlinear, show-
ing a slower rate of time increase at high ice thicknesses than at lower
ones, Figure 11 shows the total freezing time for ice thickness incre-
ments varying from 10 to 70 cm., including the time added by the heat of

20

ICE GROWTH iN cm.(Aé)

iCE GROWTH INcem.(A€)

147

12

10

St) =O (FRESH WATER)
CURVE (1) 8 *=0-l0cm.
€, == 60cm.
Sty= 35 %oo
| CURVE (2) 5 “= 0-10 cm.
al = 1 = = 60cm,
2 4 6 8 10 12 14 16
TIME (DAYS)
FIG. 9. EFFECT OF SALINITY ON RATE OF FREEZING.

LAYER DEPTH

FIG. 10. EFFECT OF HEAT OF WATER MASS ON

RATE OF FREEZING.

TIME (DAYS)

21

= 100 m.

INITIAL ICE THICKNESS = 15cm.

Berea >
Bore cee

12 G UNS ONS 4N le SmSlnSoln4 Ona 4a Ses > mSe

35

TOTAL ICE THICKNESS IN cm.

ICE GROWTH (cm)

TIME (DAYS) (T)

65

75

55

45

35

25

70
60 @
ae, ‘ ae oG:
- fs (F ,o° oP
are ° a Aa %
50 ae aD AG: Av
%
Ay Ay

16) SNOW DEPTH 8° 0
WIND VELOCITY V,= 10 mlyhr,
AIR TEMP. Tot-20°C
Equiv, TEMP, @ =-26°C
FREEZING TEMP

30 4 LS OF WATER 1y=-1.6°C
INITIAL ICE
THICKNESS £15 em.
LAYER DEPTH «100m.

20 | = 5 | aaa

10 | all

° 1S

© 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88
TOTAL TIME (DAYS)
FIG,11. EFFECT OF WATER TEMPERATURE ON THE GROWTH OF ICE
11

10

SNOW DEPTH

MEAN TEMP. OF CONVECTIVE
LAYER

Tz =0.5°C

LAYER

8 =o IcE GROWTH A&=iocm }
WIND VELOCITY V=!Omi/hr —& = INITIAL ICE. THICKNESS Ses aaa Ey Al
EQUIVALENT TEMP. 9=-25°C H = THICKNESS OF CONVEC- of
AIR TEMP To= -20°C TIVE LAYER ap

FREEZING TEMR T)=-1.6°C T = TIME ADDED BY HEAT OF Lyf

FIG.12-
OF ICE.

70 80 90 100

CONVECTIVE DEPTH, (i), (METERS)

EFFECT OF THICKNESS OF MIXED LAYER ON GROWTH

5

ine)

ICE THICKNESS (cm)

the water layer. This figure indicates, that at a mean water temperature
of -1.5°C. and for an ice thickness increment of 70 cm., the water mass
adds only approximately 18% of the total time, while at a mean water
temperature of 0.5°C. the heat of the water mass adds about 63% of the
total time,
Figure 12 shows the increase in the freezing time for 10 cm. of ice
for differing initial thicknesses ranging from 25 to 85 cm., when the
water layer increases in depth from 10 to 100 meters. At an initial
thickness of 25 cm, and a layer depth of 10 m., no appreciable time is
added to the time needed to freeze 10 om. of ice when the meteorological
conditions are as indicated on the diagram. However, when the warm layer
depth is increased to 20 m., the time increases by about 0.5 day. The
variation from this point to a layer depth of 100 m. is essentially linear.
At the 100 m. layer depth the time added for an initial thickness of 25
em, is about 10.2 days. All of these values are computed for a mean layer
temperature of 0.5°C. At the upper limit of initial ice thickness, 85 cm.
the increase of freezing time added by a layer depth of 40 meters is
practically zero; at 100-meter depth it is about 5.2 days.

F, PRACTICAL ICE THICKNESS FORECASTING

The above analysis is mainly concerned with the development of theo-
retical formulas expressing ice thickness as a function of time. In order
to take up the problem of the practical prediction of ice thickness there
are two main avenues of approach, (1) the graphical method, and (2) the
computational method.

The graphical method consists in the utilization of a diagram showing
the ice thickness as a function of time and the meteorological parameters.
For this purpose the diagrams coustituting Figure 13 are suitable. In
these diagrams the ice thickness is on the vertical seale, the air temper-
ature on the horizontal scale, and the wind velocity lines indicate the
growth curves for varying velocities. The figure is divided into three
parts for cloud coverages of 0, 50 and 100% and shows ics growth over a
period of 14 hours. Now, for example, iff it is required to know the thick-
ness of jce which will result from a temperature of ~15°C., wind velocity
of 5 m/sec., cloud cover 0%, for a period of 1 hours, the required value
taken from Figure 13a is 6.0 em. If the cloud cover had been 50% instead
of O, Figure 13a would give the required thickness as 5.5. cm., and if the
cloud cover were 100%, the thickness taken from Figure Ba would be 4.8 cm.
To determine the ice thickness for a longer period than Uy hours it is
only necessary to know the average value of the parameters over each 14-
hour period and make a cumulative sum of the thickness for integral multi-
plies of 1, hours plus the fractional part. This diagram, however, has a
definite weakness in that it does not take inte account the snow cover or
the initial thickness of the ice, both of which are important factors in
determining the rate at which ice is formed, This is a weakness inherent
in all diagrams of this type, because of the fact that it is impossible to
include:all of the important parameters on the diagram. For this reason,

(9) SSSNHOIHL 39!
2 2

Qo ee

. AIR TEMPERATURE (°F)

FIG,13. EFFECT OF CLOUDINESS AND WIND SPEED ON GROWTH OF ICE

°
>
a e
z
1

E

(%o)
D0 (KNOTS)
Rae Ss

+14

SINAC
ARNE
ANNE
FCC ONNEET:
aa eles

(WIJSSSNMDIHL 39!

°

(FREEZING TIME

14 HOURS)

EUNANIAC TEED
FERN
Ne
ore

(WO)SSANHOINL 3d!

5 a a, ao eas
AIR TEMPERATURE (°C)

FIG.13a, EFFECT OF CLOUDINESS AND WIND SPEED ON GROWTH OF ICE (FREEZING TIME

3)
°o

i4 HOURS)

then, where greater accuracy is required than is available in Figure 13a,
recourse must be made to computation. This is not a difficult process,
however, using equation (22, As pointed out previously, it is only nec-
essary to evaluate first the left and then the right sides of the equa-
tion by means of the given meteorological and oceanographic factors, and
then divide the left by the right side of the equation to find the time
expressed in units of a 24-hour day.

That this process yields accurate results can be shown by the fol-
lowing examples: The observed growth of the ice at Archangel averaged
over a l2-year period is shown in Figure 14 and the computed growth for
the same period in the same figure. The close agreement is immediately
evident. It is to be noted that the meteorological parameters used in
the computation of Figure 9 are the same as the observed values and that
the air temperature was -20°C. and the effective temperature -25°C.

Similarly, the observed ice thickness growth observed on the Denmark
Expedition from November 1, 1906, to February 8, 1907, is shown in Figure
15 and the ice growth computed by equation 22 is shown on the same figure.
Complete meteorological data including humidity, cloud cover and type of
cloud, and snow density were not available from the records, but probable,
reasonable values were used which resulted in good agreement with the
observed ice thickness data.

More recent ice observation data is that for Padloping Island showing
the observed and computed ice growth for January 1949 and January 1950.
The observed ice thickness data for this station is shown in curve 1 on
Figures 16 and 17, This station is located off Baffin Island, Davis Strait
at about 67°N. The weather data for this station were obtained from the
Padloping Island Ice Observer's Log, and the snow and ice data from the
weather summary. The computed ice growth is indicated in curve 2, The
agreement between the observed and computed ice thickness is quite close.
The weather, snow and ice data from which the computations were made are
indicated on the figures.

G. EMPIRICAL FORMULAS FOR ICE GROWIH PREDICTION
Perhaps one of the best known empirical formulas for ice accretion
as it varies with time is that of Zubov (1938). It is of the form:

; g +50f=8 Xlo ; (26)

where € = ice thickness and
To = air temperature.

This expression was obtained by the use of observed data from a particular
location with particular average values of the various parameters involved.

Comparing the results obtained from the use of this equation with
observed results obtained at Archangel, averaged for a 12-year period, the

25

ICE GROWTH IN cm. (Ahj)

80

70} 1 1

ee) ml T

neater | Sica —24.6°C
50 pets
“|| OBSERVED b=
“|—+-—} —
SPAIE(o} of
AVERAGE AIR TEMPERATURE
30} 7 |

ICE GROWTH IN-cm.(A€)

E GROWTH
ARCHANGEL, RUSSIA — 12=YEAR AVERAGE
“Th, INITIAL ICE THICKNESS
[Re INITIAL SNOW THICKNESS
COMPUTED ICE GROWTH
FROM KOLESNIKOV'S EQUATION (22)

OBSERVED

20 40 60 80 100 20 40 60 80 200 20 40 60 80 300 20 40 60 80 400
DEGREE DAYS OF FROST (°C.)

FIG.14. OBSERVED AND COMPUTED ICE GROWTH AT ARCHANGEL, U.S.S.R.

INITIAL ICE THICKNESS® 72 cm.

20 ie dis

10 i =I | [ Beil eet

j/~— COMPUTED FROM EQUATION 22+
io

(o} =—

Lo) 10 20 30 40 50 60 70 80 90 100
TIME (DAYS)

FiG.15. OBSERVED AND COMPUTED ICE GROWTH, DENMARK
EXPEDITION RECORDS, NOV.1,1906,TO FEB.8,1907.

26

TIME (DAYS)

TIME (DAYS)

32

28

24

20

16

= 3.0 Knots
+ 10°F
RH = = 82°,
Cloudiness= 0.5

DATE = 24 - 31
& 576.2'em
AG: 5.1m
8 217.8 cm
fe = 0.35
Vv, = 3.0 Knots
T 2 «10°F
RH 74°,
Cloudiness= 0.7

E ance

7-14 DATE = 14 - 21 DATE = 21 — 28
86.4 cm a (3) 291.4 cm ee > 96.5cm
5.1m AR 5.tem Qe 2 2.54 em
12.7 cm 8 16.2 em S$ = 15.2. em
0.4 Po > 0.4 fe = 0.3
= 3.0 Knots i = 3.0 Knots Wa 2 3.0 Knots

m 15'-20°C i 0 Scene 5 SeeERE

RH = 66 °/, RH 257°/, RH 245°/,
Cloudiness z 0,4 Cloudiness> 0.4 Cloudiness=: 0,4

. See:
— ee
0

ICE THICKNESS ( cm.)
FIG,16. OBSERVED AND COMPUTED ICE GROWTH AT PADLOPING
1SLAND, N.W.T., JANUARY 1949.

86 88 90

92 94 96 98 100
ICE THICKNESS (cm.)

FIG.17, OBSERVED AND COMPUTED ICE GROWTH AT PADLOPING

ISLAND, N.W.T., JANUARY

1950.
27

‘e
curves of Figure 18 were obtained. Curve 1 shows the observed time re-
quired for the accretion of ice. Curve 2 shows the ice accretion computed
by evaluating Zubov's formula with the first constant determined so that
the initial value of time was equal to the observed value. Curve 3 is the
same when the constant is determined to make the final value the same as
the final observed value. Curve 4 is found by using an average of the
two constants in curves 2 and 3. A second-degree polynomial of the form
of Zubov's equation can be determined by a statistical analysis of the
observations. An equation of this type will yield a curve with a compara-
tively close fit to the observed data, However, in all these cases the
value of the constants depends upon the average value of the parameters
involved, Thus, this expression of Zubov's as with other empirical re—
lations of this form, is valid only for observations obtained under simi-
lar conditions to those for which they are derived and is not generally
applicable to all locations and all meueore etc! and oceanographic condi-
tions. j

Another empirical equation of this type is that of Barnes (1928) »
which is of the form:

g?42¢ nh mp cae NS 8 (27)
t :

where & = iee thickness,
A, = conductivity of ice,

K =K +t (c, 42 ~C))

k = latent heat of crystallization,
freezing point of sea water,
specific heat of ice,
Cy = specific heat of sea water, ~
/, = density of ice,
f, = density of sea water,
AT, = difference in temperature between the top and bottom of the
ice, and
t = times

This expression evaluates theoretically the constant which Zubov secures
empirically. No account, however, is taken of snow thickness, which is

of paramount importance in determining the rate of accretion of ice thick-
ness.

=
=
=-
=

28

TIME (DAYS)

an Z ees a

ICE THICKNESS (cm)

FIG. 18. EMPIRICAL FORMULAE OF ZUBOV'S
TYPE COMPARED WITH OBSERVED ICE GROWTH
AT ARCHANGEL, U.S.S.R.

29

H. CONCLUSION

From the foregoing discussion it is evident that for the degree of
accuracy necessary under ordinary conditions, the use of equation (23) with
a value of taken from Table I is adequate for forecasting ice thickness
when there is a snow cover on the ice surface and when the ice has con=-
siderable initial thickness. With no snow cover and neglecting the initial
ice thickness, the graphical method of Figure 13 furnishes an adequate
solution. Only for a greater degree of accuracy is 1% necessary to make
an individual computation of © taking account the particular coefficients
of radiation, convection, evaporation, condensation, humidity, and reflection,

30

BIBLIOGRAPHY
®

ABELS, G. Measurement of the snow density at Ekaterinburg during
the winter of 1890-1891, Academia Nauk, Memoirs, vol. 19,
1892,

BARNES, H. T. Ice Engineering, Montreal: Renouf, 1928.

DEVIK, 0. M. Thermische und Dynamische Bedingungen der Eisbildung,
Wasserlaufen aug Norwegische Verhaltnisse Angewandt. Oslo,
1931,

EFIMOV, N. G. Magnitudes of the total incident radiation for
certain points of the U.S.S.R., Meteorology and Hydrology,
No. 5, 1939.

FALKENBERG, G. Absorptionskonstanten Einiger Meteorologisch
Wichtiger Korper fur Infrarote Wellen, Meteorologische
Zeitschrift, No. 334, 1928.

FRANK, H. Die Warmeabgabe Ebener Flachen an Freie Luft, Gesimdh.
dnd. 52, 1929.

KOLENSNIKOV, A. G. On the theory of ice accretion on the sea surface,
Problems of Marine Hydrological Forecasts. Leningrad, 1946.

MALMGREN, F. On the Properties of sea-ice, The Norwegian North Polar
Expedition with the MAUD, 1918-1925, vol. 1, no. 5. Bergeny
1927.

WEBER, H. Differential Gleichungen. Braunschwelg, 1910,

STEFAN, J. Uber die Theorie der Hisbildung, Isbesondere uber die
Eisbildung im Polarmeere, Akademie der Wissesnchaften, Wien,
Mathematischnaturwissenschaftliche Kiasse, Sitzungsberichte,
Band 98, Abt, 2a, 1889.

ZUBOV, N. N. Marine water and ice, Gidrometeoizdat. Moskva, 1938.

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